Properties

Label 5.227_233.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 227 \cdot 233 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$52891= 227 \cdot 233 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - x^{3} + x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.227_233.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{2} + 127 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 16 a + 1 + \left(106 a + 42\right)\cdot 131 + \left(66 a + 50\right)\cdot 131^{2} + \left(104 a + 109\right)\cdot 131^{3} + \left(76 a + 127\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 108 + 112\cdot 131 + 5\cdot 131^{2} + 120\cdot 131^{3} + 68\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 38 a + 87 + \left(76 a + 125\right)\cdot 131 + \left(81 a + 101\right)\cdot 131^{2} + \left(76 a + 12\right)\cdot 131^{3} + \left(119 a + 5\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 25 + 55\cdot 131 + 64\cdot 131^{2} + 107\cdot 131^{3} + 108\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 93 a + 108 + \left(54 a + 130\right)\cdot 131 + \left(49 a + 89\right)\cdot 131^{2} + \left(54 a + 106\right)\cdot 131^{3} + \left(11 a + 13\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 115 a + 65 + \left(24 a + 57\right)\cdot 131 + \left(64 a + 80\right)\cdot 131^{2} + \left(26 a + 67\right)\cdot 131^{3} + \left(54 a + 68\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.